Download Nuclear lattice model and the electronic configuration

Nuclear lattice model and the electronic configuration of the
chemical elements
Jozsef Garai
Department of Civil Engineering, University of Debrecen, Hungary
E-mail address: [email protected]
Abstract The fundamental organizing principle resulting in the periodic table is the nuclear
charge. Arranging the chemical elements in an increasing atomic number order, a symmetry
pattern known as the Periodic Table is detectable. The correlation between nuclear charge and the
Periodic System of the Chemical Elements indicates that the symmetry emerges from the
nucleus. Nuclear symmetry can only exist if the relative positions of the nucleons in the nucleus
are invariant. Pauli exclusion principle can also be interpreted as the nucleons should occupy a
lattice position. Based on symmetry and other indicatives face centered cubic arrangement have
been proposed for the nuclear lattice. The face centered cubic arrangement consists exclusively
tetrahedron and octahedron sub-units. Based on the higher density of the tetrahedron sub-unit,
the length of the first period and other considerations it is suggested that the nuclear structure
should be developed from a tetrahedron “seed”. Expanding this tetrahedron sub-unit and
assuming that protons and neutrons are alternate arranged then the number of the protons in the
expanded layer of the tetrahedron are identical with the main quantum numbers. The
reproduction of the angular and the magnetic momentum quantum numbers by the lattice
positions are also identical. Denser nuclear structure can be achieved if the expansion of the
tetrahedron is shifted by 90 degrees rotation at each step. The number of protons in the layers of
this “double” tetrahedron reproduces the periodicity of the chemical elements. Mathematical
equations for all of the sequences of the periodic table are derived by counting the nucleons in
the tetrahedron nuclear model. One of the important outcome of the lattice structure of the
nucleus is that the nucleus can not be considered as a point charge because the position of the
individual protons are varies and their attraction on the electrons are different. Following
classical approach it can be predicted that as long as the attraction of an added proton to the
structure is stronger than the existing ones then the captured electron joins to the valence shell of
the electrons. This condition practically is the reiteration of the Aufbau principle. When the
attraction on the captured electron is weaker than on the existing ones then a new valence
electron shell should be started. The relative strength of the attraction of the protons is described
by the distance between the proton and the charge center of the nucleus. Shorter distance
corresponds to weaker attraction and vice versa. Based on the geometry of the nuclear structure
it is shown that when a new “layer” of the nuclear structure starts then the distance between the
first proton in the new layer and the charge center of the nucleus is smaller than the distance of
the proton, which completed the preceding “layer”. Thus a new valence electron shell should
start to develop when the nuclear structure is expanded. The expansion of the double tetrahedron
FCC nuclear lattice model offers a feasible physical explanation how the nucleus affects the
electronic configuration of the chemical elements depicted by the periodic table.
Keywords Periodic System of the Chemical Elements ▪ Periodic table ▪ Electronic Structure
of the Elements ▪ Nuclear Structure ▪ Nuclear Lattice Model ▪ Sequences of the Periodic
Table ▪ Aufbau Principle ▪ Pauli Exclusion Principle
1. Introduction
1.1 Preface
In the earliest days of science researchers were arguing philosophically what was the reasonable
reason for an observed phenomenon. The majority of the contemporary scientific community
claim that these arguments are useless because they do not add anything to our understanding of
nature. The current consensus on the aim of science is that science collect facts (data) and
discern the order that exist between and amongst the various facts (e.g. Feynman, 1985).
According to this approach the mission of science is over when the phenomenon under
investigation has been described. It is left to the philosophers to answer the question why and
figure it out what is the governing physical process of the phenomenon. Quantum mechanics is a
good example of this approach ”It works, so we just have to accept it”. The consequence is that
nearly 90 years after the development of quantum theory, there is still no consensus in the
scientific community regarding the interpretation of the theory’s foundational building blocks
(Schlosshauer et al. 2013). I believe that identifying the physical process governing a natural
phenomenon is the responsibility of science. Dutailly (2013) expressed this quite well: A ”black
box” in the ”cloud” which answers rightly to our questions is not a scientific theory, if we have
no knowledge of the basis upon which it has been designed. A scientific theory should provide a
set of concepts and a formalism which can be easily and indisputably understood and used by the
workers in the field.
In this study the main unifying principle in chemistry, the periodic system of the chemical
elements (PSCE) is investigated. The aim of the study is not only the description of the
periodicity but also the understanding of the underlying physics resulting in PSCE.
1.2. The periodic pattern of the chemical elements emerged
By 1860 about 60 elements were identified, and this initiated a quest to find their systematic
arrangement. Based on similarities, Döbereiner in 1829 suggested grouping the elements into
triads. Newlands (1864) in England arranged the elements in order of increasing atomic weights
and, based on the repetition of chemical properties, proposed the “Law of Octaves”. Listing the
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elements also by mass, Mendeleev (1869; 1872) in Russia and Meyer (1870) in Germany
simultaneously proposed a 17-column arrangement with two partial periods of seven elements
each (Li-F and Na-Cl) followed by nearly complete periods (K-Br and Rb-I). Mendeleev gets
higher credit for this discovery because he published the results first; he also rearranged a few
elements out of strict mass sequence in order to fit better to the properties of their neighbors and
corrected mistakes in the values of several atomic masses. Additionally, he predicted the
existence and the properties of a few new elements by leaving empty cells in his table.
Mendeleev’s periodic table did not include the noble gasses, which were discovered later. Argon
was identified by Rayleigh in 1895/a-b. The remainder of the noble gasses were discovered by
Ramsey (1897) who positioned them in the periodic table in a new column. Anton van den
Broek (1911; 1913) suggested that the fundamental organizing principle of the table is not the
weight but rather the nuclear charge, which is equivalent with the atomic number. The extended
18-column table was slightly modified based on Moseley’s experiments (1913), and he
rearranged the table according to atomic number. The discovery of the transuranium elements
from 94-102 by Seaborg (1951) expanded the table. He also reconfigured the table by placing
the lanthanide/actinide series at the bottom. There is no “standard” or approved periodic table.
The only specific recommendation of the International Union of Pure and Applied Chemistry
(IUPAC), which is the governing body in Chemistry, is that PSCE should follow the 1 to 18
group numbering (Leigh, 2009) (Fig. 1).
1.3. Sequences of the Table
The organizing pattern of PSCE can be described by three sequences. These digital descriptions
of the table are the fundamental [Sfundamental], the periodic [SΔZ], and the atomic number [SZ]
sequences (Fig. 1) (Garai, 2008). The fundamental sequence of the table is:
[1]
The number of elements [ΔZ(n)] within the period or the length of the periods has the sequence
of
.
[2]
The atomic number or the nuclear charge of the elements [Z(n)] in a completely developed
period follows the sequence
.
3
[3]
Fig. 1. The sequences of the periodic table. The sequence derived from quantum theory is also shown. It
can be seen that the sequence derived from quantum theory does not match with the sequences of PSCE,
indicating that the theory should not be considered as a viable physical explanation for the periodicity
1.4 Pauli Exclusion Principle
In order to explain the observed light emission patterns observed in atoms Pauli postulated the
exclusion principle: “In an atom there cannot be two or more equivalent electrons for which the
values of all four quantum numbers coincide. If an electron exist in an atom for which all of
these numbers have defined values, then this state is occupied.” (Pauli, 1925; 1964). The
principle originally was postulated to electrons but it has been generalized and now includes
other particles. The Pauli Exclusion Principle is one of the corner stones of quantum physics and
thus it is at the basis of the foundation of modern physics. It is connected to spin statistics
dividing the world into fermions and bosons. Particles with half-integer spin (fermions) are
described by antisymmetric wave functions and particles with integer spin (bosons) are described
by symmetric wave function (Pauli, 1946). The rise of the periodic table is one of the important
outcome of the Pauli exclusion principle.
Pauli in his Nobel lecture in 1946 (Pauli, 1946) stated that “I was unable to give a logical
reason for the exclusion principle or to deduce it from more general assumptions. I had always
the feeling and I still have it today, that this is a deficiency.” After many decades past the
physical explanation for Pauli’s postulate is still lacking (Kaplan, 2013). Pauli exclusion
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principle is the theoretical basis for the periodic table. Thus the physical explanation for the
periodicity of the chemical elements is also lacking.
1.5 Physical Explanations
The Aufbau principle, how the electronic configuration of the atoms built up, originally was
postulated by Niels Bohr and Wolfgang Pauli. They stated that “The orbitals of lower energy are
filled in first with the electrons and only then the orbitals of high energy are filled.”. The energy
levels calculated from the hydrogenic model, or the main quantum number follows the sequence
of 2n2. Only the first two entries of this quantum number sequence is consistent with the
periodic sequence. In order to explain the periodic sequence by quantum theory the (n, l) rule or
the Madelung energy ordering (Karapetoff, 1930; Madelung, 1936) has been suggested, which
applies to neutral atoms in their ground state. The majority of general and physical chemistry
books present the symmetry of PSCE as satisfactorily explained by either the electronic structure
of the elements (Bohr, 1922; Hund, 1925; Slater, 1930; Condon, 1935; Landau & Lifschitz,
1977; Schwabl, 2001; Atkins & Atkins, 2001) or by quantum mechanics (Hartree, 1957; Fischer,
1977; Johnson, 2005). Some authors present quantum justification of the rule (Demkov and
Ostrovsky, 1971; Ostrovsky, 1981, 2001). Not all of the elements comply with the (n,l) rule;
therefore, general acceptance is lacking (Scerri 2004; Boeyens, 2008). Based on the conflicting
interpretations it can be concluded that the most important periodic sequence of the chemical
elements has not been satisfactorily explained (Scerri, 1998; Schwartz and Wang, 2010;
Boeyens, 2013). One example might be the positions of Hydrogen and Helium. Hydrogen has
one 1s electron but also one electron is needed to attain inert configuration. Thus it can be either
placed in the 1st group or in the 17th group. Based on chemical behavior, Hydrogen neither
halogen nor alkali metal but rather both. Thus the position of Hydrogen in the PSCE is
uncertain (Scerri, 2007). Helium with its 1s2 electron configuration is the other element with
uncertain position. Helium should be in the 2nd group. However, based on its chemical
properties, equivalent to inert gas, is placed into group 18. Besides Helium, the outermost
electron configuration of group 18th is 2p6. No theoretical explanation for the shift in the
electron configuration from 2p6 to 1s2 or vice versa is offered. Based on these deficiencies, it has
been suggested that quantum mechanics is unable to explain the most important aspects of the
periodic table (Scerri, 1998; Boeyens, 2008; Schwartz and Wang, 2010; Boeyens, 2013). The
substantial number of articles in the current literature (e.g. Sneath, 2000; Giunta, 2001; Kragh,
2001; Ostrovsky, 2001; Scerri 2001; Dudek et al. 2002; Baum, 2003; Dordrecht, 2003;
Ostrovsky, 2003; Moore, 2003; Scerri, 2003; Friedrich, 2004; Kibler, 2004; Schunck & Dudek,
2004; Scerri, 2004; Schwarz, 2004; Rouvray & King, 2005; Bent, 2006; Rouvray & King, 2006 ;
Wang, 2006; Restrepo & Pachon 2007; Schwarz, 2007; Weinhold & Bent, 2007; Boeyens &
Levendis, 2008; Wang & Schwartz, 2009) discussing the fundamental problems of PSCE nearly
one-and-a-half centuries after its invention, also indicates that the complete physical
understanding of the symmetry expressed by PSCE is still lacking.
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Despite the remaining outstanding questions, there is a general agreement that the elements
should be arranged in an increasing atomic number order in PSCE. Invariant symmetry relatig to
the nuclear charge can only be possible if the positions of the nucleons are invariant too.
2. Nuclear Models
The most widely accepted models for the nucleus are the shell (Mayer, 1949; Haxel et al., 1949;
Rainwater, 1950; Mayer & Jensen, 1955), the liquid drop (Bohr, 1936; Feenberg, 1947) and the
cluster (Hafstad & Teller, 1938) models. The shell model assumes a gas phase (Fermi) for the
nucleus and is able to explain the independent quantum characteristics of the nucleons. The
liquid drop model is able to explain the observed saturation properties of the nuclear forces, the
low compressibility of the nucleus, the well-defined nuclear surface, the binding energies and
most importantly the fission phenomena. The clustering of alpha particle model was deduced
from the fact that certain large nuclei emits alpha particles and the stability and the abundance of
the 4n-nuclei is significantly higher. These models are able to describe successfully certain
selected properties of the nucleus; however, none of them able to give a comprehensive
description. The basic assumptions of the models are contradictory; therefore, it is impossible
combine them and develop a hybrid model.
The assumed phase of the nucleus in these models is gas, liquid or semi-solid. None of these
phases have an invariant nucleon position; therefore, these models cannot maintain the symmetry
present in the PSCE. The preservation of symmetry requires a “solid phase” nucleus, in which
the positions of the nucleons are preserved. “Solid phase” or lattice models have not been
considered for many decades as a viable option because of the uncertainty principle and the lack
of diffraction. In the 1960s, the discovery of quarks and neutron star research satisfactorily
answered these objections and opened the door for lattice models. The first nuclear lattice
models were presented by Linus Pauling in 1965/a-c.
The lattice models can easily reproduce the various shell, liquid-drop, and cluster properties
(Cook & Dallacasa, 1987; Cook & Hayashi, 1997). Asymmetric fission and heavy-ion multifragmentation are some of the phenomena that the traditional models of nuclear structure theory
cannot explain, yet can be reproduced by lattice models (Gupta et al., 1996; 1997).
Significant effort has been made to find correlation between lattice positions and quantum
numbers with partial success for FCC structure (Pauling, 1965/d; Cook & Dallacasa, 1987). The
symmetries of the Schrödinger’s equation are also correspond to FCC geometry (Wigner, 1937).
The common features of the developed lattice models are that the protons and the neutrons have
the same size and they are alternately arranged in a closest packing array (Anagnostatos,1973;
Canuto & Chitre, 1974; Lezuo, 1974; Cook, 1976; Matsui et al., 1980; Dallacasa, 1981). These
assumptions are reasonable.
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Fig. 2. Previous lattice models of the nucleus. These investigations expanded the FCC nuclear lattice
structure spherically.
Fig. 3. The sub-units of the face centered cubic structure and their densities. The density of the FCC
structure is 0.74. The density of the sub-units, tetrahedron and octahedron are 0.7796 and 0.7209
respectively. The ratio of the tetrahedron and octahedron sub-units in the FCC structure is 2:1.
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The radii of protons and the neutrons differ only slightly (Schery et al., 1980). The same
proton and neutron magic numbers indicate same structural development for both protons and
neutrons, which is consistent with an alternate arrangement. The equal spheres will most likely
utilize the available space in the most efficient way, which is a closest packing arrangement.
Previous investigations (Pauling, 1965/d; Cook & Dallacasa, 1987), which expanded the FCC
structure spherically, had partial success in finding correspondence between lattice positions and
quantum states (Fig. 2). The structure of the closest packing arrangements consists of
tetrahedron and octahedron sub-units exclusively. The expansion of these sub-units should be
investigated first if someone wants to look for structural symmetry patterns in the FCC lattice
(Garai, 1999; 2003) (Fig. 3).
3. Tetrahedron FCC Lattice Model
Assuming that the bonding energy correlates to the nucleon density then the densest structure
should be formed preferably. The densities of the tetrahedron and octahedron units in FCC are
0.7796 and 0.7209 respectively (Garai, 2010) indicating preferential tetrahedron formation.
The first sequence of the nuclear structural development is completed by the nucleus of
Helium, which contains four nucleons. The closest packing arrangement of the four nucleons is
tetrahedron. Forming a tetrahedron sub-unit nucleus in the first completed period is an
additional support for tetrahedron nucleus formation. Calculations of potential models,
constrained by the hadron spectrum for the confinement of the relativistic quark (Goldman et al.
1988; Maltman et al. 1994) and colored quark exchange model (Robson, 1978), are also
consistent with a tetrahedron forming the He nucleus. The expansion of this tetrahedron seed of
four nucleons is investigated here.
The tetrahedron shape of equal spheres arranged in FCC packing can be formed from layers
of equilateral triangles packed in two dimensional closest packing arrangement as shown in Fig.
4/a. Starting with one sphere and increasing the length of the side of the triangles by one
additional sphere, the number of nucleons in each triangle plane will be 1, 3, 6, 10, 15, 21, 28,
36... (Fig. 4/a). Stacking these layers, the number of spheres in two consecutive layers are 4, 16,
36, 54... Assuming that protons and neutrons are alternately arranged in the lattice the number of
spheres should be divided by two. This gives the proton numbers to 2, 8, 18, 32... respectively
(Fig. 4/b). These numbers are identical with the number of possible states of the principle
quantum numbers. If the tetrahedron is expended on two faces then a shell-like structure can be
formed (Fig. 4/c), which is consistent with the physical interpretation of the principle quantum
number.
Investigating how many spheres can be accommodated in one row in the outer shell gives the
total number of spheres in one row to 4, 12, 20, 28..., which corresponds to 2, 6, 10, 14... proton
number (Fig. 5/a). These proton numbers are identical with the number of states determined by
8
the angular momentum quantum numbers corresponding to s, p, d and f orbitals. The rows in the
outer layers of the tetrahedron are one unit distance away from each other; thus the identical
agreement of the number of nucleons with the angular momentum quantum numbers is not only
in numerical agreement but also bears the same physical meaning defined by quantum theory.
Fig. 4. Representing both protons and neutrons with equal spheres and arranging them in FCC structure, the
number of protons in the outer layers of a tetrahedron formation is the same as the number of possible states
of the principle quantum numbers.
(a) The number of spheres in a two dimensional closed packing arrangement in equilateral triangles.
(b) The number of spheres in two consecutive layers of the tetrahedron formation. Assuming a protonneutron ratio of one, the outer layers of the tetrahedron contain the same number of protons as predicted by
quantum theory.
(c) The same tetrahedron formation can be developed by adding the new layers to alternate sides.
9
The number of different positions of protons in one row of the outer shell is the same as the
number of magnetic quantum numbers (Fig. 5/b). The lattice positions also reproducing the
multiplicities. Thus the number of positions in an FCC lattice are identical with the quantum
numbers if a tetrahedron seed is expanded. The lattice positions not only reproduce all of the
quantum numbers but also bear the same physical meaning.
Fig. 5. If a tetrahedron has been developed from a core tetrahedron, which contains four spheres, then the
number of protons in one layer of the outer shell of the tetrahedron is equivalent with the number of states of
the angular momentum quantum number or the corresponding sub-shell. The number of different positions
of the protons in one layer of the shell is the same as the number of magnetic quantum numbers. The red
circles represent the outer shell nucleons. The tetrahedron has four vertexes; therefore, the number of spheres
is multiplied by four.
(a) Number of spheres in one layer of a vertex of the tetrahedron.
(b) Number of the different proton positions in one layer of a vertex of a tetrahedron.
It has been assumed that higher nuclear density is preferable to lower ones. The density of
structures built in FCC arrangement can be described by the ratio of the tetrahedron and
octahedron sub-units. The higher ratio corresponds to higher nuclear density. It can be shown
that if a tetrahedron is expanded by rotating 90 degrees at each expansion then the density of this
joint or double tetrahedron is higher in comparison to a single tetrahedron. Based on the higher
density it is suggested that the initial single tetrahedron should be alternately developed by
rotating 90 degrees at each expansion of the tetrahedron. Three dimensional images of the
completed tetrahedrons corresponding to the elements He, Ne, Ar, Kr, Xe and Ra are shown in
Fig. 6/a-f. The nuclear FCC tetrahedron lattice structure expanded by rotating 90 degrees
reproduces not only the quantum numbers but also the periodicity of the chemical elements.
10
Fig. 6. 3D images show the completed nuclear structures of the noble gases. (a) Helium (b) Neon (c) Argon
(d) Krypton (e) Xenon (f) Radon
This study focuses on the relationship between the nucleus and the PSCE; therefore, the
additional characteristics of the nucleus, supporting the tetrahedron lattice model, are only listed
here.
11
● The expansion of tetrahedron into four dimensions with angles of 109.5 degrees
reproduces the original tetrahedron symmetry for every fourth nucleus. This is consistent with
the observed zero magnetic momentum for each even-even nucleus.
● The disintegration of a structure should produce fragments of its basic lattice. The
fragmentation of the tetrahedron FCC nuclear lattice is consistent with nuclear fission. The
preferred alpha decay of the nuclear structure is consistent with the disintegration of an FCC
lattice, which is built up from tetrahedron sub-units.
● The measured bulk density of the nucleus is consistent with the density of the FCC lattice
arrangement (Cook and Dallacasa,1987; Cook, 2010).
● The charge density distribution of the individual elements (Hofstadler, 1961) is also
consistent with the shape of the “double” tetrahedron FCC nuclear model.
4. Mathematical Description of the Periodic System
The proposed tetrahedron nuclear lattice model can be used to derive the sequences of the
periodic table. The three noticeable sequences in the periodic table are the fundamental {1, 2, 2,
3, 3, 4, 4...}, the periodic {2, 8, 8, 18, 18, 32, 32...}, and the atomic number{2, 10, 18, 36, 54, 86,
118 ...} sequences (Fig. 1).
4.1 Fundamental Sequence
The relationship between the periods (n) and the sequence numbers (m) can be described as:
[4]
4.2 Periodic Sequence
The number of nucleons in the
layer of a tetrahedron can be calculated by the triangular
number [Tr(k)] (Abramowitz & Stegun, 1964; Beiler, 1964) (Fig. 4/a)
.
[5]
In each structural step of its development, the tetrahedron is expanded by one layer in two
directions (Fig. 4/c) giving the relationship between the tetrahedron layers and the sequence
numbers as:
.
[6]
12
The number of nucleons in the outer shell of the tetrahedron [Tr(m)] is the sum of the two
consecutive triangular numbers.
[7]
The number of charges in the completely developed shell is
[8]
Substituting the sequence number from Eq. 4 gives:
[9]
which is the same formula given by Tomkeieff (1951; 1954).
4.3 Atomic Number Sequence
A formula giving the total number of charges in the nucleus with completely developed shells
can be derived in a similar manner. The total number of nucleons in a tetrahedron with k layers
can be determined by its tetrahedral number [Th(k)] (Abramowitz & Stegun, 1964; Conway,
1996)
[10]
Substituting the sequence number from Eq. (6) gives the number of nucleons in a tetrahedron for
sequence (m) as:
[11]
The tetrahedron nucleus is developed by alternately expanding the tetrahedrons rotated by 90
degrees (Fig. 4/d). The number of nucleons in the tetrahedron is
.
[12]
The formula
[13]
can be used to generate 0 for odd periods and 1 for even number periods, and
[14]
Equation (12) can be rewritten then as
13
[15]
The number of charges in the nucleus in a completely developed sequence is
.
[16]
Combining Eqs. (4), (7), (11), (12), (15) and (16) gives the number of nuclear charges for any
period. The atomic number sequence of the periodic table can be described then as:
.
[17]
Substituting m from Eq. (4) gives the atomic number sequence (Garai, 2008)
[18]
Any physical models proposed for the explanation of PSCE has to explain and reproduce the
sequences of the table.
For comparison the sequence of the Principle Quantum Numbers [PQN(n)] is given as:
.
[19]
The first two entries are the same in both the sequence of the principle quantum number and the
sequence of the length of the period. The rest of the entries in these sequences shows similarity
but they are not identical. The sequence of the principle quantum number can be given as:
.
[20]
This formula [Eq. (20)] emerging from quantum theory does not agree with any of the sequences
of the periodic table [Eqs.(4, 9, 18)] indicating that quantum mechanics can not give a viable
explanation for the sequences of the periodic table.
5. Pauli Exclusion Principle
The chemical properties of the elements is defined by the electronic structure of the outermost or
valence shell. The identity of an atom, including its electronic structure is determined by the
nucleus or more specifically by the number of protons (Broek, 1911, 1913). The interaction
between the atomic particles can be described as:
Strong Force Effects
Neutrons
⇔
Protons
Electrostatic Effects
⇒
14
Electrons.
The negatively charged electrons in an atom are captured by the electrostatic attraction of the
positively charged protons. The electrostatic attraction between the two differently charged
particles is described by an inverse square law. The attraction between a proton and a captured
electron is the function of the distance between the two charges. The energy of a captured
electron in its ground state depends on the distance between the two particles. Electrons with
different energies in their ground states should be separated by different distances from the
protons capturing them. This distance must be invariable otherwise the electronic structure of
the atom would not be stable. In order to maintain the configuration of the protons and the
distances between the protons and the captured electrons the position of the protons must be
invariant. This condition is fulfilled by a nuclear lattice. The lattice structure of the nucleus
ensures that if a lattice is occupied then another proton can not have the same position. Thus the
nuclear lattice model offers a feasible physical explanation for the Pauli Exclusion Principle,
which requires that the protons in the nucleus cannot co-exist in the same location. It is
concluded that in order to comply with the Pauli Exclusion Principle the protons in the nucleus
should be arranged in a lattice.
6. Physical Explanation for the Periodicity of the Electronic Structure
The most important feature in the development of the electronic structure of the chemical
elements is the {2, 8, 8, 18, 18, 32, 32...} pattern which represents the number of elements in
each period of the periodic table or the number of electrons in each completed shell. The
periodicity pattern for the ground state neutral atoms and for the positive atomic ions is different
(Brakel, 2000; Goudsmith & Richards, 1964). The most simple manifestation of periodicity, the
ground state neutral atoms, are investigated here. The periodic sequence is described by
equation 9, and it is attributed to the number of protons in the outer shell of the tetrahedron
nuclear structure. It will be investigated that how protons in the nuclear lattice can affect the
electronic structure of the elements.
The interaction between the proton and electron is the result of the exchange of virtual
photons. Based on the energies of the electrons in the atoms virtual photon or quantum
electrodynamic treatment is not necessary and the classical approach is sufficient to describe the
interaction. The classical approach is supported by the success of the Bohr model, which
correctly describes the orbit of electron in the Hydrogen atom (Bohr, 1913).
One of the outcome of the nuclear lattice model is that the nucleus can not be considered as a
point charge. The lattice position of a given proton must be taken into consideration when the
attraction on an electron is calculated. The relative attractive force [Fe-P(Z)] of a proton [P(Z)] on
an electron [e(Z)] is characterized by the distance between the charge center of the nucleus and
the proton [dNCC-P(Z)]
15
Fe-P(Z) = f (dNCC-P(Z)).
[21]
where Z is the corresponding atomic number defined as follows. The first proton in the
Hydrogen nucleus is P(1), the next proton added into the structure to form the nucleus of Helium
is P(2) and the last proton added to the nuclear structure of element Z-1 to form the nucleus of
element Z is P(Z). The electrons are labeled accordingly. Thus electron(1) is captured by P(1)
and electron(Z) is captured by P(Z). Depending on the lattice positions of proton(Z) the
attraction can be either stronger or weaker than the attraction of proton(Z-1). The relative
attraction of proton Z-1 and Z can be defined by comparing the distances between the charge
center of the nucleus and the position of the proton:
then
[22]
or vice versa. In order to overcome the repulsion force of the existing valence electrons, the
relative attractive force on the captured electron(Z) must be stronger than the relative attraction
force on the electrons already occupying the valence shell. Thus the following inequality must
be satisfied:
.
⇔
[23]
This inequality is consistent with the Aufbau principle, which states that orbitals with the
lowest energy are filled first. When this condition [Eq. 23] is not satisfied then it is predicted
that the electron(Z) is unable to join to the valence shell and starts to form a new valence shell
outside of the existing one. Thus a new electron shell starts to form when the condition [Eq. 24]
is satisfied.
FP-e(Z) < FP-e(Z-1) ⇔ dP-NCC(Z) < dP-NCC(Z-1)
[24]
Based on stability considerations the first proton in the new nuclear shell should be
positioned at the middle of the face of the tetrahedron and this new outer layer of the structure
should develop towards the edges. It can be seen by visual inspection that in a given nuclear
shell equation 23 or the Aufbau principle is satisfied (Fig. 7/a).
The distance of the last proton completing a shell [n] and the distance of the first proton in
the new shell [n+1] is calculated from the nuclear charge center. It is assumed that the position
of the last proton in layer n is at the vertex of the tetrahedron and the position of the first proton
in layer n+1 is at the surface, closest to the charge center (Fig. 7/b). It is also assumed that the
charge center of the nucleus coincides with the mass center. For the completed structures of the
tetrahedron He, Ne and “double” tetrahedron Ar, Xe and Ra nucleus, the mass center coincides
with the charge centers. When a new nuclear shell is started by adding a proton, then the charge
center shifts towards the proton resulting in smaller distance between the charge center and the
proton than the distance between the proton and the mass center. Thus using the mass center
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instead of the charge center is a conservative estimate on the inequality of dP-NCC(Z) < dP-NCC
(Z-1).
Fig. 7. Schematic figure in 2D showing the geometry of the nucleus.
(a) The distance between the the nuclear charge center (NCC) and the proton increases as the protons occupy
the lattice positions in the same layer.
(b) When a new layer starts to build up in the nucleus, the proton is closer towards to the nuclear charge
center than the proton completing the previous shell. Thus proton [P(Z+1)] starting the period of n+1 has
weaker attraction on the captured electron than the proton [P(Z)] completing the period n has. The captured
electron(Z+1) is unable to overcome the repulsion of the existing shell because of its weaker attraction and
starts to built up a new valence shell outside of the existing one.
The distance between the first proton in the shell and the mass center is calculated first. For
simplification the length of the edge of the basic tetrahedron is one unit. Thus the diameter of
the nucleons is also one unit. The height of a tetrahedron [h∆] with unit length is
.
[25]
Assuming a unit mass for the nucleons the distance between the vertex of the tetrahedron and the
mass center is:
[26]
The distance between the face of the tetrahedron and the mass center is then:
[27]
The tetrahedron is expanded by one-one layers on both sides; thus the length of the side of
the tetrahedron is increased by two units in each periods (Fig. 7). The distance between the
center of a sphere (nucleon lattice position), placed on the surface of the tetrahedron, and the
mass center in period n+1 is then
17
[28]
where n is the number of period. The distance between a sphere placed at the vertex of the
tetrahedron in period n can be calculated then as:
[29]
It can be seen that
dΔvertex-MC(n) > dΔsurface-MC(n+1)
[30]
where n = 2, 3, 4, 5... Arranging the nucleons in an FCC lattice and building up a tetrahedron
inequality in equation 24 is fullfilled when a new layer in the nuclear structure starts to form
(Fig. 7/b). Thus
dP-NCC(Z=2) > dP-NCC(Z=3) ● dP-NCC(Z=10) > dP-NCC(Z=11) ● dP-NCC(Z=18) > dP-NCC(Z=19)
dP-NCC(Z=36) > dP-NCC(Z=37) ● dP-NCC(Z=54) > dP-NCC(Z=55) ● dP-NCC(Z=86) > dP-NCC(Z=87).
[31]
The remaining rest of the pairs of the elements ([Z]⇔[Z+1]) satisfy the relationship of equation
23 and
.
Resulting from the inequalities of Eq. 31 the relative attraction force on the newly captured
electron is weaker than the force on the electrons in the existing shell leads to the formation of a
new electron shell for the following elements:
[32]
Based on the presented geometrical consideration it is suggested that the cyclical structural
development of the nuclear structure results in an interruption of the development of valence
shell because the relative attraction of the proton on the captured electron becomes weaker when
a new layer starts to built in the nucleus (Garai, 2011). The cycles of the weaker relative
attractions predicted from the geometry of the nuclear structure are identical with the length of
the periods in the periodic table. It is concluded that the electronic configuration of the chemical
elements is the consequence of the structural development of the nuclear lattice.
7. Conclusions
The symmetry pattern of the periodic system of the chemical elements emerges from the nuclear
charge. This invariant pattern can be maintained only if the positions of the nucleons are also
invariant. Thus the nucleons should form a lattice structure. The lattice arrangement of the
nucleons is consistent with the Pauli exclusion principle and offers a feasible physical
18
explanation for the exclusion principle. Representing protons and neutrons with equal spheres,
arranging them alternately in an FCC lattice, and developing a tetrahedron, rotating by 90
degrees at each expansion results in an identical symmetry pattern expressed by the PSCE. The
lattice positions of the nucleons not only reproduces the quantum numbers but they also bear the
same physical meaning indicating that the presented model should be considered as a credible
physical explanation for quantum theory.
Based on the presented nuclear lattice model mathematical solutions for the sequences of the
periodic table are derived. Investigating the structural development of the nucleus it is shown
that the periodicity of the electronic structure is the natural outcome of the nuclear lattice
geometry. The presented tetrahedron lattice model is the first nucleus model which is able to
reproduces all of the sequences of the periodic table, offers a credible physical explanation for
the identical symmetry of the nucleus and the electronic structure, for the Pauli exclusion
principle and for the Aufbau principle.
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